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The three-loop splitting functions \(P_{q g}^{(2)}\) and \(P_{g g}^{(2, \operatorname{N}_{\operatorname{F}})}\). (English) Zbl 1373.81370
Summary: We calculate the unpolarized twist-2 three-loop splitting functions \(P_{q g}^{(2)}(x)\) and \(P_{g g}^{(2, \mathrm{N}_{\mathrm{F}})}(x)\) and the associated anomalous dimensions using massive three-loop operator matrix elements. While we calculate \(P_{g g}^{(2, \mathrm{N}_{\mathrm{F}})}(x)\) directly, \(P_{q g}^{(2)}(x)\) is computed from 1200 even moments, without any structural prejudice, using a hierarchy of recurrences obtained for the corresponding operator matrix element. The largest recurrence to be solved is of order 12 and degree 191. We confirm results in the foregoing literature.

MSC:
81V05 Strong interaction, including quantum chromodynamics
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
65Q30 Numerical aspects of recurrence relations
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[1] Accardi, A., Eur. Phys. J. C, 76, 8, 471, (2016)
[2] Alekhin, S.; Blümlein, J.; Moch, S.; Placakyte, R., Parton distribution functions, \(\alpha_s\) and heavy-quark masses for LHC run II
[3] Bethke, S.; Moch, S.; Alekhin, S.; Blümlein, J.; Moch, S. O., High precision fundamental constants at the TeV scale, Mod. Phys. Lett. A, 31, 25, (2016)
[4] Boer, D.; Abelleira Fernandez, J. L., Gluons and the quark sea at high energies: distributions, polarization, tomography, J. Phys. G, 39, (2012)
[5] Moch, S.; Vermaseren, J. A.M.; Vogt, A., Nucl. Phys. B, 688, 101, (2004)
[6] Vogt, A.; Moch, S.; Vermaseren, J. A.M., Nucl. Phys. B, 691, 129, (2004)
[7] Larin, S. A.; van Ritbergen, T.; Vermaseren, J. A.M.; Larin, S. A.; Nogueira, P.; van Ritbergen, T.; Vermaseren, J. A.M.; Retey, A.; Vermaseren, J. A.M.; Blümlein, J.; Vermaseren, J. A.M.; Bagaev, A. A.; Bednyakov, A. V.; Pikelner, A. F.; Velizhanin, V. N.; Velizhanin, V. N., Nucl. Phys. B, Nucl. Phys. B, Nucl. Phys. B, Phys. Lett. B, Phys. Lett. B, Nucl. Phys. B, 864, 113, (2012)
[8] Blümlein, J.; Klein, S.; Tödtli, B., Phys. Rev. D, 80, (2009)
[9] Bierenbaum, I.; Blümlein, J.; Klein, S., Nucl. Phys. B, 820, 417, (2009)
[10] Ablinger, J.; Behring, A.; Blümlein, J.; De Freitas, A.; von Manteuffel, A.; Schneider, C., Comput. Phys. Commun., 202, 33, (2016)
[11] Blümlein, J.; Schneider, C., Phys. Lett. B, 771, 31, (2017)
[12] Ablinger, J.; Blümlein, J.; De Freitas, A.; Hasselhuhn, A.; von Manteuffel, A.; Round, M.; Schneider, C.; Wißbrock, F., Nucl. Phys. B, 882, 263, (2014)
[13] Ablinger, J.; Behring, A.; Blümlein, J.; De Freitas, A.; Hasselhuhn, A.; von Manteuffel, A.; Round, M.; Schneider, C.; Wißbrock, F., Nucl. Phys. B, 886, 733, (2014)
[14] Ablinger, J.; Behring, A.; Blümlein, J.; De Freitas, A.; von Manteuffel, A.; Schneider, C., Nucl. Phys. B, 890, 48, (2014)
[15] Ablinger, J.; Blümlein, J.; De Freitas, A.; Hasselhuhn, A.; Schneider, C.; Wißbrock, F., Three loop massive operator matrix elements and asymptotic Wilson coefficients with two different masses, (2015), DESY 14-019
[16] Blümlein, J., Comput. Phys. Commun., 159, 19, (2004)
[17] Vermaseren, J. A.M., Int. J. Mod. Phys. A, 14, 2037, (1999)
[18] Blümlein, J.; Kurth, S., Phys. Rev. D, 60, (1999)
[19] Remiddi, E.; Vermaseren, J. A.M., Int. J. Mod. Phys. A, 15, 725, (2000)
[20] Buza, M.; Matiounine, Y.; Smith, J.; Migneron, R.; van Neerven, W. L., Nucl. Phys. B, 472, 611, (1996)
[21] Bierenbaum, I.; Blümlein, J.; Klein, S., Nucl. Phys. B, 780, 40, (2007)
[22] Bierenbaum, I.; Blümlein, J.; Klein, S.; Schneider, C., Nucl. Phys. B, 803, 1, (2008)
[23] Buza, M.; Matiounine, Y.; Smith, J.; van Neerven, W. L., Eur. Phys. J. C, 1, 301, (1998)
[24] Bierenbaum, I.; Blümlein, J.; Klein, S., Phys. Lett. B, 672, 401, (2009)
[25] Nogueira, P., J. Comput. Phys., 105, 279, (1993)
[26] van Ritbergen, T.; Schellekens, A. N.; Vermaseren, J. A.M., Int. J. Mod. Phys. A, 14, 41, (1999)
[27] Studerus, C., Comput. Phys. Commun., 181, 1293, (2010)
[28] von Manteuffel, A.; Studerus, C., Reduze 2 - distributed Feynman integral reduction
[29] Lewis, R. H., Computer algebra system
[30] Bauer, C. W.; Frink, A.; Kreckel, R., J. Symb. Comput., 33, 1, (2000), cs/0004015 [cs-sc]
[31] Kauers, M., Guessing handbook, (2009), JKU Linz, Technical Report RISC 09-07
[32] Schneider, C., Sémin. Lothar. Comb., 56, 1, (2007), B56b
[33] Schneider, C., Simplifying multiple sums in difference fields, (Schneider, C.; Blümlein, J., Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions, Texts and Monographs in Symbolic Computation, (2013), Springer Wien), 325 · Zbl 1315.68294
[34] Ablinger, J.; Ablinger, J., A computer algebra toolbox for harmonic sums related to particle physics, diploma thesis, (2009), J. Kepler University Linz, p. 019
[35] Ablinger, J., Computer algebra algorithms for special functions in particle physics, (2012), J. Kepler University Linz, Ph.D. Thesis
[36] Ablinger, J.; Blümlein, J.; Schneider, C., J. Math. Phys., 52, (2011)
[37] Ablinger, J.; Blümlein, J.; Schneider, C., J. Math. Phys., 54, (2013)
[38] Ablinger, J.; Blümlein, J.; Raab, C. G.; Schneider, C., J. Math. Phys., 55, (2014)
[39] Karr, M., J. ACM, 28, 305, (1981)
[40] Schneider, C., Symbolic summation in difference fields, (2001), Ph.D. Thesis RISC, Johannes Kepler University, Linz Technical Report 01-17
[41] Schneider, C.; Schneider, C.; Schneider, C., Difference equations in πσ-extensions, An. Univ. Vest. Timis., Ser. Mat.-Inform., J. Differ. Equ. Appl., Appl. Algebra Eng. Commun. Comput., 16, 1, (2005) · Zbl 1101.39001
[42] Schneider, C., J. Algebra Appl., 6, 415, (2007)
[43] Schneider, C., (Carey, A.; Ellwood, D.; Paycha, S.; Rosenberg, S., Motives, Quantum Field Theory, and Pseudodifferential Operators, Clay Mathematics Proceedings, vol. 12, (2010), Amer. Math. Soc), 285, (2010)
[44] Schneider, C., Ann. Comb., 14, (2010)
[45] Schneider, C., (Gutierrez, J.; Schicho, J.; Weimann, M., Computer Algebra and Polynomials, Applications of Algebra and Number Theory, Lecture Notes in Computer Science, vol. 8942, (2015)), 157, arXiv:13077887 [cs.SC]
[46] Schneider, C.; Schneider, C., J. Symb. Comput., J. Symb. Comput., J. Symb. Comput., 80, 616, (2017)
[47] Sage · Zbl 1312.68206
[48] Kauers, M.; Jaroschek, M.; Johansson, F., (Gutierrez, J.; Schicho, J.; Weimann, Josef M., Computer Algebra and Polynomials, Lecture Notes in Computer Science, vol. 8942, (2015), Springer Berlin), 105
[49] Blümlein, J.; Kauers, M.; Klein, S.; Schneider, C., Comput. Phys. Commun., 180, 2143, (2009)
[50] Gross, D. J.; Wilczek, F., Phys. Rev. D, 9, 980, (1974)
[51] Georgi, H.; Politzer, H. D., Phys. Rev. D, 9, 416, (1974)
[52] Floratos, E. G.; Ross, D. A.; Sachrajda, C. T., Nucl. Phys. B, 152, 493, (1979)
[53] Gonzalez-Arroyo, A.; Lopez, C., Nucl. Phys. B, 166, 429, (1980)
[54] Furmanski, W.; Petronzio, R., Phys. Lett. B, 97, 437, (1980)
[55] Hamberg, R.; van Neerven, W. L., Nucl. Phys. B, 379, 143, (1992)
[56] Ellis, R. K.; Vogelsang, W.
[57] Moch, S.; Vermaseren, J. A.M., Nucl. Phys. B, 573, 853, (2000)
[58] Ablinger, J.; Blümlein, J.; Klein, S.; Schneider, C.; Wißbrock, F., Nucl. Phys. B, 844, 26, (2011)
[59] Klein, F.; Bailey, W. N.; Appell, P.; Kampé de Fériet, J.; Appell, P.; Kampé de Fériet, J.; Exton, H.; Exton, H.; Srivastava, H. M.; Karlsson, P. W.; Schlosser, M. J., Multiple Gaussian hypergeometric series, (Schneider, C.; Blümlein, J., Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions, (2013), Springer Wien), vol. 39, 305, (1985), Ellis Horwood Chicester
[60] Slater, L. J., Generalized hypergeometric functions, (1966), Cambridge University Press Cambridge · Zbl 0135.28101
[61] Barnes, E. W.; Barnes, E. W.; Mellin, H., Proc. Lond. Math. Soc. (2), Q. J. Math., Math. Ann., 68, 305, (1910)
[62] Czakon, M.; Smirnov, A. V.; Smirnov, V. A., Comput. Phys. Commun., Eur. Phys. J. C, 62, 445, (2009)
[63] Ablinger, J.; Blümlein, J.; Klein, S.; Schneider, C.; Blümlein, J.; Hasselhuhn, A.; Schneider, C.; Schneider, C.; Schneider, C., Nucl. Phys., Proc. Suppl., Comput.-Algebra-Rd.br., J. Phys. Conf. Ser., 523, 8, (2014), PoS (RADCOR 2011) 032
[64] Kotikov, A. V.; Caffo, M.; Czyz, H.; Laporta, S.; Remiddi, E.; Caffo, M.; Czyz, H.; Laporta, S.; Remiddi, E.; Gehrmann, T.; Remiddi, E.; Kotikov, A. V.; Kotikov, A. V.; Kotikov, A. V.; Henn, J. M., Phys. Lett. B, Acta Phys. Pol. B, Nuovo Cimento A, Nucl. Phys. B, Theor. Math. Phys., Phys. Part. Nucl., Phys. Rev. Lett., 110, 25, 374, (2013), in: D. Diakonov (Ed.), Subtleties in quantum field theory, p. 150
[65] Gerhold, S., Uncoupling systems of linear ore operator equations, (2002), J. Kepler University Linz, (the package OreSys has been tuned for our specific calculations, by the last author)
[66] Almkvist, G.; Zeilberger, D.; Apagodu, M.; Zeilberger, D., J. Symb. Comput., Adv. Appl. Math. (Special Regev Issue), 37, 139, (2006)
[67] Ablinger, J.; Blümlein, J.; De Freitas, A.; Hasselhuhn, A.; von Manteuffel, A.; Round, M.; Schneider, C., Nucl. Phys., 885, 280, (2014)
[68] Blümlein, J.; Hasselhuhn, A.; Klein, S.; Schneider, C., Nucl. Phys. B, 866, 196, (2013)
[69] Bennett, J. F.; Gracey, J. A., Nucl. Phys. B, 517, 241, (1998)
[70] Brown, F., Commun. Math. Phys., 287, 925, (2009)
[71] Ablinger, J.; Blümlein, J.; Raab, C.; Schneider, C.; Wißbrock, F., Nucl. Phys. B, 885, 409, (2014)
[72] Panzer, E., Comput. Phys. Commun., 188, 148, (2015)
[73] Vermaseren, J. A.M., New features of FORM · Zbl 1344.65050
[74] Steinhauser, M., Comput. Phys. Commun., 134, 335, (2001)
[75] Klein, S. W.G., Mellin moments of heavy flavor contributions to \(F_2(x, Q^2)\) at NNLO, (2009), TU Dortmund, PhD Thesis
[76] Itzykson, C.; Zuber, J. B., Quantum field theory, International Series in Pure and Applied Physics, (1980), McGraw-Hill New York · Zbl 0453.05035
[77] Smirnov, V. A., Feynman integral calculus, (2006), Springer Berlin
[78] Bogner, C.; Weinzierl, S., Int. J. Mod. Phys. A, 25, 2585, (2010)
[79] Tarasov, O. V., Phys. Rev. D, 54, 6479, (1996)
[80] Lee, R. N., Nucl. Phys. B, 830, 474, (2010)
[81] G. Green, Essay on the Mathematical Theory of Electricity and Magnetism, Nottingham, 1828 (Green Papers, pp. 1-115).
[82] Broadhurst, D. J., Z. Phys. C, 54, 599, (1992)
[83] Bekavac, S.; Grozin, A. G.; Seidel, D.; Smirnov, V. A., Nucl. Phys. B, 819, 183, (2009)
[84] Grigo, J.; Hoff, J.; Marquard, P.; Steinhauser, M., Nucl. Phys. B, 864, 580, (2012)
[85] Schröder, Y.; Luthe, T.; Schröder, Y., Five loop massive tadpoles, presentation at loops and legs in QFT, (2016), PoS (LL2016) 074
[86] Danilevskiĭ, A.; Barkatou, M. A.; Zürcher, B.; Bronstein, M.; Petkovšek, M.; Abramov, S. A.; Zima, E. V.; Bostan, A.; Chyzak, F.; de Panafieu, E., Rationale normalformen von pseudo-linearen abbildungen, (ISSAC 2013, Boston, (2013)), 157, 1, 16, (1996), ETH Zürich, and references therein
[87] Kamke, E., Differentialgleichungen: Lösungsmethoden und Lösungen, (1967), Geest & Portig Leipzig · Zbl 0026.31801
[88] Blümlein, J.; Broadhurst, D. J.; Vermaseren, J. A.M., Comput. Phys. Commun., 181, 582, (2010)
[89] Collins, J. C.; Vermaseren, J. A.M., Axodraw version 2 · Zbl 1114.68598
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